System and method for generating a low discrepancy curve on an abstract surface

Abstract

A system and method for generating a curve, such as a Low Discrepancy Curve, on a surface, such as an abstract surface with a Riemannian metric. The system may comprise a computer which includes a CPU and a memory medium which is operable to store one or more programs executable by the CPU to perform the method. The method may: 1) parameterize the surface; 2) select a curve, such as a Low Discrepancy Curve, in a parameter space, for example, a simple space such as a unit square; 3) re-parameterize the surface, for example, re-parameterize the surface such that a ratio of line and area elements of the surface based on a Riemannian metric is constant; and 4) map the curve onto the surface using the re-parameterization. The method may also generate output comprising information regarding the mapped curve, for example, displaying the mapped curve on a display device.

1. A method for generating a curve on a surface, comprising:
 selecting a parameterization of the surface, wherein the parameterization corresponds to a parameter space;  
 selecting a first curve in the parameter space, wherein the first curve is a Low Discrepancy Curve in the parameter space;  
 determining a re-parameterization of the surface based on a metric of the surface; and  
 mapping the first curve in the parameter space onto the surface based on the re-parameterization, to generate the curve, wherein the generated curve is a Low Discrepancy Curve on the surface;  
 wherein the curve is useable in analyzing the surface.  
 
   
   
     2. The method of  claim 1 , further comprising:
 generating output comprising the generated curve on the surface.  
 
   
   
     3. The method of  claim 1 , wherein the parameter space comprises a 2D rectangle. 
   
   
     4. The method of  claim 1 , wherein the parameter space comprises a unit n-dimensional cube, wherein the dimensionality n is greater than 4. 
   
   
     5. The method of  claim 1 , wherein the parameter space comprises one of a unit square, a unit cube, or a unit hyper-cube. 
   
   
     6. The method of  claim 1 , wherein the metric of the surface is a Riemannian metric. 
   
   
     7. The method of  claim 1 , wherein the surface comprises a space with dimensionality greater than two. 
   
   
     8. The method of  claim 1 ,
 wherein the surface comprises a six-dimensional space comprising three position degrees of freedom and three orientation degrees of freedom for an object; and  
 wherein the generated curve comprises a scan path useable in scanning the space.  
 
   
   
     9. The method of  claim 8 , further comprising:
 scanning the space, wherein said scanning the space is performed to determine an optimum position and orientation for the object.  
 
   
   
     10. The method of  claim 1 , wherein the parameter space comprises a 2D rectangle. 
   
   
     11. The method of  claim 1 , wherein the parameter space comprises one of a unit square, a unit cube, or a unit hyper-cube. 
   
   
     12. The method of  claim 1 , wherein the parameter space comprises a unit n-dimensional cube, wherein the dimensionality n is greater than 4. 
   
   
     13. The method of  claim 1 , wherein the metric of the surface is a Riemannian metric. 
   
   
     14. The method of  claim 1 , wherein the surface comprises a space with dimensionality greater than two. 
   
   
     15. A method for generating a curve on a surface of an object, comprising:
 selecting a parameterization of the surface, wherein the parameterization corresponds to a parameter space;  
 selecting a first curve in the parameter space, wherein the first curve is a Low Discrepancy Curve in the parameter space;  
 determining a re-parameterization of the surface based on a metric of the surface; and  
 mapping the first curve in the parameter space onto the surface based on the re-parameterization, to generate the curve, wherein the generated curve is a Low Discrepancy Curve on the surface;  
 wherein the curve is useable in analyzing the object.  
 
   
   
     16. The method of  claim 15 , further comprising:
 generating output comprising the generated curve on the surface.  
 
   
   
     17. The method of  claim 15 , further comprising:
 analyzing the object using the curve.  
 
   
   
     18. The method of  claim 11 , further comprising:
 determining a property of the object using the curve.  
 
   
   
     19. A memory medium which is operable to store program instructions for generating a curve on a surface, wherein said program instructions are executable to perform:
 storing a parameterization of the surface, wherein the parameterization corresponds to a parameter space;  
 storing a first curve in the parameter space, wherein the first curve is a Low Discrepancy Curve in the parameter space;  
 determining a re-parameterization of the surface based on a metric of the surface; and  
 mapping the first curve in the parameter space onto the surface based on the re-parameterization, to generate the curve, wherein the generated curve is a Low Discrepacy Curve on the surface;  
 wherein the curve is useable in analyzing the surface.  
 
   
   
     20. The memory medium of  claim 19 , wherein the program instructions are further executable to perform:
 generating output comprising the generated curve on the surface.  
 
   
   
     21. The memory medium of  claim 19 , wherein the parameter space comprises one of a 2D rectangle, a unit square, a unit cube, or a unit hyper-cube. 
   
   
     22. The memory medium of  claim 19 , wherein the parameter space comprises a unit n-dimensional cube, wherein the dimensionality n is greater than 4. 
   
   
     23. The memory medium of  claim 19 , wherein the metric of the surface is a Riemannian metric. 
   
   
     24. The memory medium of  claim 19 , wherein the surface comprises a space with dimensionality greater than two. 
   
   
     25. The memory medium of  claim 19 ,
 wherein the surface comprises a surface of an object;  
 wherein the curve comprises a scan path useable to scan the object;  
 wherein the program instructions are further executable to perform:  
 scanning the object to determine one or more properties of the object.  
 
   
   
     26. The memory medium of  claim 19 ,
 wherein the surface comprises a six-dimensional space comprising three position degrees of freedom and three orientation degrees of freedom for an object;  
 wherein the generated curve comprises a scan path useable in scanning the six-dimensional space;  
 wherein the program instructions are further executable to perform:  
 scanning the six-dimensional space to determine an optimum position and orientation of the object.  
 
   
   
     27. A system for generating a curve on a surface, comprising:
 a CPU; and  
 a memory medium which is operable to store one or more software programs;  
 wherein said CPU is operable to execute said one or more software programs to perform:  
 selecting a parameterization of the surface, wherein the parameterization corresponds to a parameter space;  
 selecting a first curve in the parameter space, wherein the first curve is a Low Discrepancy Curve in the parameter space;  
 determining a re-parameterization of the surface based on a metric of the surface; and  
 mapping the first curve in the parameter space onto the surface based on the re-parameterization, to generate the curve, wherein the generated curve is a Low Discrepancy Curve on the surface;  
 wherein the curve is useable in analyzing the surface.  
 
   
   
     28. The system of  claim 27 , wherein the CPU is further operable to execute said one or more software programs to perform:
 generating output comprising the generated curve on the surface.  
 
   
   
     29. The system of  claim 27 , wherein the parameter space comprises one of a 2D rectangle, a unit square, a unit cube, a unit hyper-cube, or a unit n-dimensional cube, wherein the dimensionality n is greater than 4. 
   
   
     30. The system of  claim 27 , wherein the metric of the surface is a Riemannian metric. 
   
   
     31. The system of  claim 27 , wherein the surface comprises a space with dimensionality greater than two. 
   
   
     32. A method for generating a curve on a surface, comprising:
 selecting a parameterization of the surface, wherein the surface comprises a space with dimensionality greater than two, and wherein the parameterization corresponds to a parameter space;  
 selecting a first curve in the parameter space;  
 determining a re-parameterization of the surface based on a metric of the surface; and  
 mapping the first curve in the parameter space onto the surface based on the re-parameterization, to generate the curve;  
 wherein the curve is useable in analyzing the surface.  
 
   
   
     33. A method for generating a curve on a surface of an object, comprising:
 selecting a parameterization of the surface, wherein the surface comprises a space with dimensionality greater than two, wherein the parameterization corresponds to a parameter space;  
 selecting a first curve in the parameter space;  
 determining a re-parameterization of the surface based on a metric of the surface; and  
 mapping the first curve in the parameter space onto the surface based on the re-parameterization, to generate the curve;  
 wherein the curve is useable in analyzing the object.  
 
   
   
     34. A memory medium which is operable to store program instructions for generating a curve on a surface, wherein said program instructions are executable to perform:
 storing a parameterization of the surface, wherein the surface comprises a space with dimensionality greater than two, wherein the parameterization corresponds to a parameter space;  
 storing a first curve in the parameter space;  
 determining a re-parameterization of the surface based on a metric of the surface; and  
 mapping the first curve in the parameter space onto the surface based on the re-parameterization, to generate the curve;  
 wherein the curve is useable in analyzing the surface.  
 
   
   
     35. A system for generating a curve on a surface, comprising:
 a CPU; and  
 a memory medium which is operable to store one or more software programs;  
 wherein said CPU is operable to execute said one or more software programs to perform:  
 selecting a parameterization of the surface, wherein the surface comprises a space with dimensionality greater than two, wherein the parameterization corresponds to a parameter space;  
 selecting a first curve in the parameter space;  
 determining a re-parameterization of the surface based on a metric of the surface; and  
 mapping the first curve in the parameter space onto the surface based on the re-parameterization, to generate the curve;  
 wherein the curve is useable in analyzing the surface.